Non–equilibrium statistics of a reduced model for energy ...deville/research/DMPTV-CPAM-March2005.pdf · Non–equilibrium statistics of a reduced model for energy transfer in waves

Non–equilibrium statistics of a reduced modelfor energy transfer in waves

R. E. LEE DEVILLE

Courant Institute

PAUL A. MILEWSKI

University of Wisconsin, Madison

RICARDO J. PIGNOL

U. Nacional del Sur, Bahıa Blanca, Argentina

ESTEBAN G. TABAK

Courant Institute

AND ERIC VANDEN-EIJNDEN

Courant Institute

Abstract

We study energy transfer in a “resonant duet” — a resonant quartet where symme-tries support a reduced subsystem with only two degrees of freedom — where onemode is forced by white noise and the other is damped. We consider a physicallymotivated family of nonlinear damping forms, and investigate their effect on the dy-namics of the system. A variety of statistical steady-states arise in different parameterregimes, including intermittent bursting phases, non–equilibrium states highly con-strained by slaving among amplitudes and phases, and Gaussian and non-Gaussianquasi–equilibrium regimes. All of this can be understood analytically using asymptotictechniques for stochastic differential equations.

1 Introduction

Many wave systems in nature are best described in terms of Fouriermodes, and the nonlinearities of the dynamics correspond to energy ex-change amongst these modes. Often the dynamics is conservative in alarge range of length scales (the inertial range), with forcing and dissipa-tion acting only over a more restricted range. For example, ocean surfacewaves are thought to be initiated by the wind at small (capillary) scales,with the subsequent dynamics transferring the energy to longer scales.The main dissipation mechanism is wave breaking, which usually acts onmuch longer (gravity) waves that intermittently remove energy from thewave system.

For dispersive wave systems, such as surface ocean waves and inter-nal waves in the atmosphere and ocean, the conservative energy transferoccurs mostly through resonant sets, typically triads or quartets. Weak

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0027 (2003)c© 2003 John Wiley & Sons, Inc. CCC 0010–3640/98/000001-27

ENERGY TRANSFER IN WAVES 2

turbulence (or wave turbulence) theory describes this weakly nonlinearconservative energy cascade through the resonant sets. The theory yieldskinetic equations (where the energy exchange is through wave “collisions”)for the evolution of the Fourier spectrum. Sometimes, special power lawsolutions of these equations with a prescribed flux of energy can be found[6, 2, 1, 7, 11].

In idealized systems where the damping acts on infinitely small or largelength-scales (the so-called inviscid limit), it is believed that the form ofdamping does not affect the spectrum. Away from this limit, however,the form of the forcing may affect the solution. For instance, developers ofGeneral Circulation Models for the Atmosphere and Ocean are well awareof the sensitivity of their models to the form of the parameterization ofdamping used. The purpose of the present work is to understand theeffect of more realistic, nonlinear dissipation in the context of a simplemodel amenable to detailed analysis. To this end, we generalize the duetsystem introduced in [9]. The forms of dissipation that we consider includethose consistent with wave breaking and therefore may prove useful inparameterizing wave breaking in more complex scenarios.

As in [9], the model emerges from isolating one resonant quartet in ageneral dispersive system, adding white noise forcing and dissipation, andthen further reducing the system to just two complex degrees of freedom,by exploiting a symmetry. The reason to model the forcing through whitenoise is that this permits a complete control of the energy input, andhence also of the rate of energy transfer through the system, when it isin a statistically stationary state.

The resulting duet system has many analogies to more complex dis-persive systems, but is of low enough dimension that it is quite amenableto numerical simulation. Furthermore, it is simple enough that it canbe understood almost completely using theoretical tools. In spite of itssimplicity, this model exhibits a rich variety of behaviors. In particular,it contains both Gaussian and non-Gaussian quasi–equilibrium states, inaddition to states far from equilibrium, dominated by a maximally effi-cient energy transfer, with most of the dynamics slaved to the evolutionof one single mode.

The paper is organized as follows.

In Section 2, we state the duet model we study in the paper, and showhow it is related to a typical nonlinear PDE. In Section 3, we study someof the elementary statistical properties of the duet model. In Section 4 wedescribe a numerical study of the duet for various parameter values and


show the existence of several parameter regimes. In Sections 5 and 6 weanalyze the dynamical equations in two different asymptotic regimes andobtain the (sometimes approximate) distributions for the system. Finally,in Section 7 we summarize and suggest further work.

2 The duet system

We consider in this paper the resonant duet

(2.1)

{

ia1 = 2γa1a22 + σW ,

ia2 = 2γa21a2 − iνa2 |a2|β ,

where a1, a2 are two complex amplitudes, γ > 0, σ > 0, ν > 0, β ≥ 0 areconstants, and W is a complex Weiner process. This is a prototypicalmodel for energy transfer; we have two coupled modes, and we force oneand damp the other with a view towards determining what proportionof the total energy remains in each mode. The parameter β allows thedamping to be nonlinear.

This is a generalization of the system studied in [9], which describedthe case β = 0, corresponding, for instance, to a fluid’s viscosity. Thepurpose of introducing a more general β is to understand the role of non-linearity in the dissipation mechanism, and as we show below this canlead to radically different qualitative behavior. There are several moti-vations to study this generalization. From a technical perspective, it isinteresting to find physical systems that have equilibrium invariant mea-sures which are non-Gaussian. From a physical perspective, dissipationin fluid problems is often brought about by nonlinear wave dynamics.

2.1 Relation to the nonlinear Schrodinger equation

As in [9], we start with a one-dimensional partial differential equation ofthe form

(2.2) i∂Ψ

∂t= LΨ + γ |Ψ|2 Ψ + forcing and damping,

where L is a Hermitian linear operator with symbol L = ωk. In theinertial range, this system behaves in a Hamiltonian manner as

i∂Ψ

∂t=∂H

∂Ψ, H =

∫

dk ωk|Ψ(k)|2 +γ

2

∫

dx |Ψ(x)|4 .


Our goal here is to find a simple subsystem of (2.2) in which we canunderstand the mechanisms of energy transfer. First, we consider a singleresonant quartet, i.e. a set of four wavenumbers kj such that the resonantconditions

k1 + k4 = k2 + k3,

ωk1+ ωk4

= ωk2+ ωk3

,

are satisfied. If we then excite these four modes with O(ǫ) amplitudes,then the dynamics of Ψkj

can be approximated, up to leading order, by

Ψkj(t) = ǫaj(τ)e

−i(ωkj−2ǫγm)t

,

with τ = ǫ2t, m =∑

k |ak|2 (the mass of the system), and the aj satisfythe resonant equations (see e.g. [5])

(2.3)

ia1 = 2γa4a2a3 − γa1 |a1|2 ,ia2 = 2γa3a1a4 − γa2 |a2|2 ,ia3 = 2γa2a1a4 − γa3 |a3|2 ,ia4 = 2γa1a2a3 − γa4 |a4|2 .

The last term in each equation comes from the “self-interaction” of eachmode with itself (because of the trivial relation that kj + kj = kj + kj),and we will actually drop these terms in the sequel. This simplifies thecalculations but does not change them qualitatively (cf. [9] where theseterms are retained). We now drop the self-interaction terms in (2.3),noting that it retains its Hamiltonian structure, and obtain

H = 2γ(a1a2a3a4 + a1a3a3a4).

These equations satisfy the “Manley-Rowe” relations

d |a1|2dt

=d |a4|2dt

= −d |a2|2dt

= −d |a3|2dt

,

from which follows the conservation of mass m, momentum p, and linearenergy e, given as

(2.4) m =∑

j

|aj| , p =∑

j

kj |aj |2 , e =∑

j

ωkj|aj |2 .

With the fourth conserved quantity being the Hamiltonian itself, thismakes this system integrable. This system can actually be solved analyt-ically [5] but we do not need that here.


Since we want to study energy transfer in (2.3), we consider the gen-eralization

ia1 = 2γa4a2a3 + σW1(t),

ia2 = 2γa3a1a4 − iνa2 |a2|β ,ia3 = 2γa2a1a4 − iνa3 |a3|β ,ia4 = 2γa1a2a3 + σW4(t),

(2.5)

where W1 and W4 are white noise. One reason for using white-noiseforcing is the following: if we were to use deterministic forcing, the firstoscillator could reach equilibrium with the forcing by a frequency de-tuning, and no more energy would be added to the system. This is notpossible with stochastic forcing, and in the case of white noise the amountof energy entered into the system is exactly controllable.

Once we add white-noise forcing, we must damp the system some-where, or the energy would diverge. Furthermore, once we force onemode (say a1) then we must force a4 with white noise, and of the sameamplitude, to hope to have a statistical steady-state. (Otherwise, the

Manley-Rowe relations would imply that⟨

|a1|2 − |a4|2⟩

would diverge.)

There is no reason other than symmetry to choose the damping strengthsto be equal, but we do that here and exploit this symmetry.

Now, consider (2.5) with W1 = W4 and initial data so that a1(0) =a4(0), and a2(0) = a3(0). Then

a1(t) = a4(t), a2(t) = a3(t), for all t,

and (2.5) reduces to (2.1). The duet system is not a model for two inter-acting modes in a PDE such as (2.2), but it represents the reduction ofquartet dynamics to an invariant submanifold.

2.2 Relation to wave breaking

(2.1) with nonlinear damping (β 6= 0) is also reminiscent of wave break-ing. In particular, wave breaking is naturally parameterized through anonlinear dissipation of the form proposed above, with β = 1. To seethis, consider the prototypical model for wave breaking, i.e. the inviscidBurgers equation

ut +(

12u

2)

x= 0 .


Here the energy e = 12u

2 satisfies

et +

(

2√

2

3e3/2

)

x

= 0 .

The energy dissipation integrated over space is proportional to the sumof the jumps of e3/2 at shocks. In a profile of fixed shape – say a sawtoothfor the asymptotic behavior of Burgers in a periodic domain– which canbe parameterized by the amplitude a of its first Fourier mode, the energye is proportional to |a|2, while the dissipation rate et is proportional to|a|3. This is the result of our nonlinear dissipation when the parameter βis set to one.

3 Elementary properties of the duet system

The system (2.1) is Hamiltonian with

H = γ(a21a

22 + a2

1a22),

and the mass, momentum, and linear energy in (2.4) all become the samequantity, which we will now call the energy, denoted

(3.1) E = |a1|2 + |a2|2 .

It will be convenient to use the variables ak = ρkeiθk , ξk = ρ2

k, giving

(3.2)

dξ1 = (4γξ1ξ2 sin(2θ) + σ2) dt+ σ√

2ξ1 dW1,

dξ2 = (−4γξ1ξ2 sin(2θ) − 2νξβ/2+12 ) dt,

dθ = 2γ(ξ2 − ξ1) cos(2θ) dt +σ√2ξ1

dWθ,

where W1 and Wθ are independent white noises. It is straightforward tosee that for any fixed β this system has a single non-dimensional param-eter, which we define here as

D = γ−1σ2β−2

β+2 ν4

β+2 .

Now, in the interests of deciding which scaling limits are appropriatebelow, we will formally compute the expected value of the size of the twoamplitudes ξ1, ξ2.

First, we calculate

d

dt〈ξ1〉 = 4γ 〈ξ1ξ2 sin(2θ)〉 + σ2,


d

dt〈ξ2〉 = −4γ 〈ξ1ξ2 sin(2θ)〉 − 2ν

⟨

ξβ/2+12

⟩

,

where 〈·〉 denotes the expectation with respect to the noise. Adding thesegives

d

dt〈E〉 = σ2 − 2ν

⟨

ξβ/2+12

⟩

,

and thus if there is a steady-state, it must be true that

⟨

ξβ/2+12

⟩

=σ2

2ν.

For the case of linear damping (β = 0) this gives an exact bound for theaverage energy of a2. If β > 0, then this is an upper bound, as usingJensen’s inequality gives

(3.3) 〈ξ2〉 ≤⟨

ξβ/2+12

⟩2

β+2 =

(

σ2

2ν

) 2

β+2

.

On the other hand, we can also consider the equation

d ln ξ2 = −4γξ1 sin(2θ) − 2νξβ/22 .

After averaging, and again assuming the existence of a steady-state, weobtain

〈ξ1 sin(2θ)〉 =−ν2γ

⟨

ξβ/22

⟩

,

and using the fact that 〈ξ1〉 ≥ |〈ξ1 sin(2θ)〉|, we have

(3.4) 〈ξ1〉 ≥ν

2γ

⟨

ξβ/22

⟩

.

4 Numerics and predictions

We used the standard first-order Euler scheme throughout to simulatethe equations here. In various cases, we sometimes simulated the originalequations (3.2), and at other times different scaled versions of the equa-tions, for example (6.1) were typically used for large D values, and (5.1)were typically used for small D values. We will state in each case belowwhich equations we simulated. Furthermore, in all numerical simulationsfor β = 0, we actually simulated ln ξ2 instead of ξ2. As we will see fromthe numerics and analysis in this paper, in this β = 0 regime, ξ2 tends tobecome very close to 0: in fact, ln(1/ξ2) reaches O(D) frequently duringmany realizations.

We will analytically explain all of the observations of this section inSections 5 and 6 below.


4.1 Scaling of amplitudes of ξ1, ξ2 with D

Figures 4.1 and 4.2 show a series of of experiments concerning which of thebounds (3.3),(3.4) are saturated in which parameter regimes. Figure 4.1corresponds the system (3.2) with β = 0, σ = γ = 1, and various ν inthe range 0.2 to 8.0. For these choices of parameters, D = ν2, so we areplotting D values from 0.04 to 64.0. To create this graph, we simulatedbetween 10 and 25 realizations of the system from t = 0 to t = 105, and wecalculated the time and ensemble average starting at t = 104 of 〈ξ1〉 and〈ξ2〉. We took timesteps ranging from ∆t = 10−3 to ∆t = 10−5. As we willsee from the analysis below, the system becomes stiff for some values ofD and we took smaller timesteps and larger ensembles when appropriate.We have plotted 〈ξ1〉 with circles and 〈ξ2〉 with stars, and plotted thebounds (3.3) and (3.4) with solid lines. The bounds (3.3),(3.4) hold, andfurthermore (3.3) is exact as predicted. Also, the inequality (3.4) seemsto get closer to an equality as D gets larger. Furthermore, as D becomessmall, the amplitudes ξ1 and ξ2 approach each other.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

3

3.5

4

4.5

D

<|a

1,2|2 >

Figure 4.1. 〈ξ1〉 (circles) and 〈ξ2〉 (asterisks) as functions of D = ν2/σ2γfor β = 0, with (3.4),(3.3) plotted as solid lines.

Figure 4.2 corresponds to the system with β = 5, σ = γ = 1, and


0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

D

<ξ 1>

, <

ξ 2>

Figure 4.2. 〈ξ1〉 (circles) and 〈ξ2〉 (asterisks) as functions of D =γ−1σ6/7ν4/7 for β = 5, with (3.3),(3.4) plotted as solid lines. Note thatfor β 6= 0, we need to calculate 〈ξ2〉 before we can draw the curve whichis the lower bound for 〈ξ1〉.

various ν in the range 0.1 to 3000. In this case we simulated the equa-tions directly with ensemble sizes, timesteps, and time domains similarto the β = 0 case. For these choices of parameters, D = ν4/7, so we areplotting D values from near 1/4 to near 100. Again, one sees that thebounds (3.3),(3.4) hold. However, notice that the bounds are not as sharpin this case. Again, for small D, the amplitudes are nearly equal.

4.2 Pathwise dynamics

In all of the following numerical simulations, we simulated rescaled ver-sions of the equations ((6.1) for large D, and (5.1) for small D). Again,for β = 0 we also simulated ln ξ2 instead of ξ2 directly.

Figure 4.3 shows realizations of the system for various D and β. Asone can see from the two pictures in the bottom row, for small D thetwo modes oscillate around each other in much the same way as in theunforced system. For large D, however, the scenario is quite different,and even depends sensitively on the value of β. In the case of large D


5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

ξ 1, ξ 2

26 26.2 26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 280

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

ξ 1, ξ

2

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 60

0.5

1

1.5

2

2.5

3

3.5

4

t

ξ 1, ξ 2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.5

1

1.5

2

2.5

t

ξ 1, ξ 2

Figure 4.3. Realizations of (6.1) for various values of D and β. Ineach case, the blue curve is ξ1 and the green curve is ξ2. The leftcolumn correspond to β = 0 while the right column corresponds toβ = 5, and the first row correspond to D = 25 while the bottomrow corresponds to D = 0.1.

but linear damping, the second oscillator stays pinned to 0 for most ofthe time, undergoing brief outbursts intermittently. On the other hand,for large D and β > 0, the amplitudes of a1 and a2 are locked.

4.3 Equilibrium probability densities

Figure 4.4 show the (numerically-calculated) densities of ξ1, ξ2 for variousvalues of D and β. For small D, one sees that the two oscillators are quitesimilar for either choice of β. There is one difference, in that for β = 0,the measures are indistinguishable from Gaussian, while for β > 0, themeasures are definitely non-Gaussian. For large D, the two oscillators actquite differently. We see that for the case of β = 0, ξ1 is close to Gaussian,but the measure for ξ2 is quite different, having a long tail. For the case


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−5

10−4

10−3

10−2

10−1

100

101

ξ1, ξ

2

p(ξ 1),

p

(ξ2)

0 0.2 0.4 0.6 0.8 1 1.2 1.410

−2

10−1

100

101

ξ1, ξ

2

p(ξ 1),

p(ξ

2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−3

10−2

10−1

100

101

ξ1, ξ

2

p(ξ 1),

p(ξ

2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−4

10−3

10−2

10−1

100

101

ξ1, ξ

2

p(ξ 1),

p(ξ

2)

Figure 4.4. Numerically-computed densities of (6.1) for various val-ues of D and β. The left column correspond to β = 0 while theright column corresponds to β = 5, and the first row correspond toD = 25 while the bottom row corresponds to D = 0.1. Where thegraph includes a descending line, this is a graph of an exponential.

of β > 0, we see that neither measure is Gaussian.

5 D ≪ 1 – The quasi–equilibrium regime

Combining (3.3) with the numerical observation (see figures 4.1, 4.2) thatthe average energy of a1 and a2 are the same for small D suggests thescalings

ξj = σ4

β+2 ν−2

β+2 ξj,

t = σ−

2ββ+2ν

− 2

β+2 t.


This scaling can be obtained by assuming that (3.3) is saturated and thatthe ξj are of the same size. This gives us

(5.1)

dξ1 = 4D−1ξ1ξ2 sin(2θ) dt + 1 dt +√

2ξ1dW1,

dξ2 = −4D−1ξ1ξ2 sin(2θ) dt − 2ξβ+2

2

2 dt,

dθ = 2D−1(ξ2 − ξ1) cos(2θ) dt +1√2ξ1

dWθ,

where we have dropped tildes. Also, notice that every term representingHamiltonian dynamics has a D−1 in front of it, and all of the terms arisingfrom forcing and damping are of order 1.

What we show below is that there exist functions Tα, ψ (see (A.6) fordefinitions) so that the dynamics of (5.1) can be approximated by

(5.2)

{

dE = 2(1 − Eβ/2+1Tβ/2+1(h)) dt +√E dW1 +

√

h2/E dW2,

dh = −2Eβ/2hTβ/2(h) dt +√

(4ψ(h) − h)h/E dW2,

where h = 2H/E2. In particular, T0 = 2 and T1 = 1, so if β = 0, we getthe simpler SDE

dE = 2(1 − E) dt +√E dW1 +

√

h2/E dW2,

dh = −4hdt +√

(4ψ(h) − h)h/E dW2.

This represents a dimensional reduction of the original system. Wehave only reduced the original system (2.1) by two degrees of freedom,but this formulation has the advantage that it has no small parameter.In the D ≪ 1 limit, (5.1) are a stiff set of equations. From a numericalpoint of view, this technique would represent a clear savings in the smallD limit. This viewpoint is entirely analogous to the reduction to the slowmanifold in singular perturbation theory for deterministic ODE.

5.1 Consequences of (5.2)

Converting (5.2) to the appropriate Fokker-Planck equation gives

Ft = (2(Eβ/2+1Tβ/2+1(h) − 1)F )E + (Eβ/2hTβ/2(h)F )h

+ (EF )EE + (2hF )Eh + (4hE−1ψ(h)F )hh.(5.3)

The steady-state distribution is any F which satisfies

0 = (2(Eβ/2+1Tβ/2+1(h) − 1)F )E + (Eβ/2hTβ/2(h)F )h

+ (EF )EE + (2hF )Eh + (4hE−1ψ(h)F )hh.(5.4)


on the domain 0 ≤ E ≤ ∞,−1 ≤ h ≤ 1, with Dirichlet boundary condi-tions.

In certain cases, we can solve (5.4) exactly. We first simplify by cal-culating the marginal distribution in E and h. We integrate (5.4) in h,obtaining

(5.5) 2(Cβ/2+1Eβ/2+1 − 1)FE + (EF )EE = 0,

where

F =

∫ 1

−1dhF (E,h), Cα =

∫ 1

−1dh Tα(h).

The only solution of (5.5) which is bounded at infinity is

(5.6) F (E) = E exp(−kE1+β/2),

with k = 4Cβ/2+1/(β + 2). We note that this agrees with the resultsobtained in [9] and [10], where a similar analysis was done using theimplicit assumption that the distribution only depends upon E. Also, onecan see from figure 5.1 that this agrees quite well, for various choices of β,with the numerically calculated distribution. Note that this distributionis not Gaussian for β > 0.

Furthermore, in the case where β = 0, it turns out that the solutionfor F in (5.3) actually separates, and the PDE can be solved explicitly toobtain

F (E,h) = Ee−E exp(−χ(h)/2)

hψ(h),

where χ(h) =∫ h0 dh

′ ψ(h′). The authors cannot find such a nice separationof variables for the β > 0 case.

5.2 Derivation of (5.2) and (5.3)

The corresponding generator of the diffusion in (5.1) is

1

DLH + LFD,

where

LH = 4ξ1ξ2 sin(2θ)(∂ξ1 − ∂ξ2) + 2(ξ2 − ξ1) cos(2θ)∂θ,(5.7)

LFD = ∂ξ1 + ξ1∂2ξ1ξ1 − 2ξ

β+2

2

2 ∂ξ2 +1

4ξ1∂2

θ .(5.8)


0 0.5 1 1.510

−4

10−3

10−2

10−1

100

ξ1

p(ξ 1)

D=0.1, β=0.20

0 0.5 1 1.510

−4

10−3

10−2

10−1

100

ξ1

p(ξ 1)

D=0.1, β=0.50

0 0.5 1 1.510

−5

100

ξ1

p(ξ 1)

D=0.1, β=1.00

0 0.5 1 1.510

−8

10−6

10−4

10−2

100

ξ1

p(ξ 1)

D=0.1, β=2.00

0 0.5 110

−3

10−2

10−1

100

ξ1

p(ξ 1)

D=0.1, β=5.00

0 0.5 110

−6

10−4

10−2

100

102

ξ1

p(ξ 1)

D=0.1, β=10.00

Figure 5.1. A pdf of the energy of (6.1) for small D and various valuesof β. The circles are the numerically determined distributions, and theline is the prediction made in (5.6), with k determined numerically.

We are considering D ≪ 1, so that the LH term dominates. First notethat LH is skew. This is because it is written in the form LH = F · ∇,where F is divergence-free, and thus L∗

H = −(F · ∇) + divF = −LH .By inspection, one can see that the null space of LH (and also L∗

H forthat matter) is made up of functions of H and E. This makes sense, sinceit says exactly that any positive function of the conserved quantities canbe used to construct an invariant measure for the unforced system.

We write

(5.9) ∂tf − L∗f = ∂tf − 1

DL∗

Hf − L∗FDf = 0.

Let us assume that f can be developed in a series as

f = f0 +Df1 +D2f2 + · · ·and plugging this into (5.9) gives

L∗Hf0 = 0,

L∗Hf1 = ∂tf0 − L∗

FDf0.


The first term tells us that f0 is a function of H and E and we willdenote this F (H,E). The second equation gives a compatibility conditionof the form

∂tF − L∗FDF ∈ R(L∗

H) = N (LH)⊥.

This means that for any function G ∈ N (LH) (i.e. any function G =G(H,E)), we must have

∫

dξ1dξ2dθ (∂tF + L∗FDF )G = 0.

We can integrate by parts to write this as

(5.10)

∫

FtG+ FLFDG = 0,

where, again, G is any test function of E and H. We stress that weare not interested in determining f1 or any other term in the asymptoticexpansion; we are only trying to determine the form of f0. What wewill do in (5.10) is rewrite the FLFDG integral so that it is in H andE coordinates, then average over the remaining coordinate to obtain anoperator which depends solely on H and E. The way we do this is firstadd two integrals and delta functions, as in

∫ ∞

0dE

∫ E2/2

−E2/2dH

∫ 2π

0dθ

∫ ∞

0dξ1

∫ ∞

0dξ2

δ(H(ξ1, ξ2, θ) − H)δ(E(ξ1, ξ2, θ) − E)F (H,E)LFD(H,E, ξ1, ξ2, θ).

(5.11)

We use the two delta functions to remove two dξ integrals, and then dothe dθ integral explicitly (see Appendix A for details) to obtain

(5.12)

∫ ∞

0dE

∫ E2/2

−E2/2dHJ(H, E)F (H, E)φ(H, E),

where

J(H, E) =2

E√

1 + aK

√

1 − a

1 + a

,

φ(H, E) = (2 − 2Tβ/2+1(H, E))GE − HTβ/2(H, E)GH

+ EGEE + 2HGHE + 4EHψ(H, E)GHH ,


with a = |2H/E2|, K is the elliptic K-function, and Tα and ψ are definedbelow in (A.6). Although the expression Tα is rather complex, it is worthnoting here that T0 = 2 and T1 = E, so that for β = 0 the expression forφ(H, E) is much nicer, namely

2(1 − E)GE − 2HGH + EGEE + 2HGEH + 4EHψGHH .

Since J really represents the Jacobian of changing variables toH,E, wewant to understand the evolution of X = JF . We plug (5.12) into (5.10),integrate by parts, and note that G is arbitrary, to obtain

Xt = (2(Tβ/2+1 − 1)X)E + (HTβ/2X)H

+ (EX)EE + (2HX)EH + (2EHψX)HH .(5.13)

Further making the change h = 2H/E2 and writing F forX gives (5.3).This is of course equivalent to the SDE in (5.2).

6 D ≫ 1, The non–equilibrium regime

In contrast to the small D asymptotics of section 5, we will see here thatthere is a vast qualitative difference between the case where β = 0 andwhere β > 0, and we will deal with these two cases separately below. Inboth cases, however, we will do the scaling

ξ1 = γ−1σ2β

β+2ν2

β+2 ξ1, ξ2 = σ4

β+2ν−2

β+2 ξ2, t = γ−1σ−4

β+2 ν2

β+2 t.

This is obtained by assuming that the bounds (3.3) and (3.4) are satu-rated. The scaling gives us (after dropping tildes)

(6.1)

dξ1 = 4ξ1ξ2 sin(2θ) dt + dt+√

2ξ1 dW1,

dξ2 = D(−4ξ1ξ2 sin(2θ) − 2ξβ/2+12 )dt,

dθ = 2(ξ2 −Dξ1) cos(2θ) dt + 1√2ξ1dWθ.

6.1 Nonlinear damping, β > 0

The β > 0 case turns out to be the more straightforward of the two cases.Let us rewrite the equation for ξ2 in (6.1) as

dξ2 = −2Dξ2(2 sin(2θ)ξ1 + ξβ/22 ).

We write α(t) = sin(2θ), and for now think of this as a control parameterwhich varies in time in some specified way. For any fixed α, this equation


is strongly stable to the curve ξ2 = (−2αξ1)2/β (by strongly stable we

mean it has a large negative eigenvalue). Furthermore, if ξ2 ≪ Dξ1,the equation for θ in (6.1) is strongly stable to the value of θ for whichcos(2θ) = 0 and sin(2θ) = −1, namely θ = 3π/4.

If we happen to choose an initial condition with ξ2 = O(D), then thedynamics of ξ2 will pull the solution rapidly into a neighborhood of themanifold

ξ2 = (−2αξ1)2/β .

Once we are there, the value of θ locks to θ = 3π/4 and thus α locks to−1, and this causes ξ2 to lock to the manifold ξ2 = (2ξ1)

2/β , and then ξ2stays O(1).

We stress that although there are regions of phase space for which thereis not phase- and amplitude-locking, this is a self-correcting phenomenonsince the deterministic part of the vector field causes the system to leavethese areas quickly. With the locking, the equation for ξ1 in (6.1) becomes

dξ1 = (−22β+2

β ξ1+ 2

β

1 + 1) dt +√

2ξ1dW,

and it is straightforward to calculate that the steady-state distributionfor this equation is

(6.2) f(ξ1) = Ce−kξ1+2/β1 ,

where in fact k = 22β+2

β β/(β + 2). Thus the steady-state distribution forthe full system (6.1) is

f(ξ1)δ(ξ2 − (2ξ1)2/β)δ(θ − 3π/4).

In figures 6.1 and 6.2, we see that the numerics agree with these predic-tions. We mention that it is an implicit assumption in the above argumentthat D be chosen sufficiently large, and the minimum such choice of Dis β-dependent. As β → 0, one will need to choose larger values of D toensure phase- and amplitude-locking. It is an interesting observation thatthe distribution for ξ1 is not Gaussian for any value of β > 0.

6.2 Linear damping, β = 0

The case β = 0 is somewhat more complicated. If we consider (6.1) withβ = 0, we note that we might expect phase-locking of θ to occur, atleast for most regions of the phase space, but now there is certainly no


0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

7

ξ1

ξ 2

β = 0.500000

0 0.2 0.4 0.6 0.8 10

1

2

3

4

ξ1

ξ 2

β = 1.000000

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

ξ1

ξ 2

β = 2.000000

0 0.5 1 1.50

0.5

1

1.5

2

ξ1

ξ 2

β = 5.000000

Figure 6.1. In each graph, the curve is ξ2 = (2ξ1)2/β and the dots

are the solution of (6.1) at times t = 1000 + 25k, D = 250.

mechanism which locks ξ2 to ξ1. The situation is further complicatedbecause the θ does not stay locked for typical trajectories of the system.

We will show that the dynamics in the β = 0 regime are of two phaseswhich switch back and forth intermittently. The first phase is a quiescentone in which ξ2 is pinned very near zero and ξ1 undergoes a random walk.The second phase is characterized by a burst in the energy of ξ2. We alsowill show that the quiescent phase is essentially unaffected in its durationand quality by D, although it is affected by noise, and the bursting phaseis essentially unaffected by the noise, but is affected greatly by D. In fact,as we show below, the length, in time, of the bursts are O(lnD/D) andtheir amplitude is O(D).

Let us consider the system where β = 0, namely

(6.3)

dξ1 = (4 sin(2θ)ξ1ξ2 + 1) dt +√

2ξ1 dW1,

dξ2 = −2Dξ2(2 sin(2θ)ξ1 + 1)dt,

dθ = 2(ξ2 −Dξ1) cos(2θ) dt + (√

2ξ1)−1dWθ.

We will define the “quiescent” phase to be that in which ξ2 < 1, and


0 0.5 1 1.50

0.5

1

1.5

2

ξ1

p(ξ 1)

D=250, �eta=0.20

0 0.5 1 1.50

0.5

1

1.5

2

ξ1

p(ξ 1)

D=250, �eta=0.50

0 0.5 1 1.50

0.5

1

1.5

2

ξ1

p(ξ 1)

D=250, �eta=1.00

0 0.5 1 1.50

0.5

1

1.5

2

2.5

ξ1

p(ξ 1)

D=250, �eta=2.00

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

ξ1

p(ξ 1)

D=250, �eta=5.00

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

ξ1

p(ξ 1)

D=250, �eta=10.00

Figure 6.2. In each graph, the curve is the prediction given in (6.2)and the dots are the numerically computed distribution of (6.1).

the “bursting” phase to be that in which ξ2 > 1. The 1 is arbitrary andchosen only for definiteness. When we are in the quiescent stage, it is easyto see that θ is locked to θ = 3π/4, and since it has a multiplier of O(D),one can see that its variance is O(1/D) since ξ1 is always of O(1). In thisphase we can simplify the system, using the substitution sin(2θ) = −1,to obtain

dξ1 = (1 − 4ξ1ξ2) dt+√

2ξ1dW,

dξ2 = 2D(2ξ1 − 1)ξ2 dt.(6.4)

The ξ2 equation is linear, and we see by inspection that it always changingrapidly. It will be rapidly shrinking if ξ1 < 1

2 and rapidly growing ifξ1 > 1

2 . Consider an initial condition (ξ1, ξ2) = (A, 1) with A < 1/2.Then ξ2 moves toward zero rapidly. Effectively, the equation for ξ1 doesnot then depend on ξ2, and it is easy to see that it is a random walk witha positive drift.

If we think of ξ1(t) as a parameter forcing the equation for ξ2 in (6.4),


we get

ξ2 = exp

[

−2D

∫ t

0ds (2ξ1(s) − 1)

]

.

In particular, the quiescent stage will last until the value of t for which

0 =

∫ t

0ds (2ξ1(s) − 1).

This does not depend on D. In essence, the effect of large D in thequiescent regime is to pin ξ2 to 0 more strongly when ξ1 < 1/2, and haveit come back to 1 more quickly when ξ1 > 1/2, but this effect cancels out,and D will not affect the length of the quiescent stage. In particular, themean value 〈ξ1〉 takes when this phase ends is 1−A, and this is a quantityindependent of D.

In summary, as long as D is sufficiently large, the length of time thequiescent phase lasts is independent of D, and the values that ξ1 takesduring this phase are also independent of D. On the other hand, thevalues that ξ2 takes during this phase depend on D greatly, in that theygo like e−Dt for the first half of this phase (while ξ1 < 1/2) and then growlike eDt in the second half (while ξ1 > 1/2).

Now we consider the bursting phase. We will see that the assumptionthat θ is locked in this phase is not valid. In fact, let us assume thatθ is locked to θ = 3π/4, and show that this is not a self-consistent as-sumption. Consider the system (6.4). For definiteness, let us consider aninitial condition (B, 1) with B > 1/2. It is easy to see that the burstingphase will not be greatly affected by the noise, since during the burstthe deterministic vector field is everywhere O(D). Thus we consider thedeterministic equation

dξ1 = −4ξ1ξ2 dt,

dξ2 = 2D(2ξ1 − 1)ξ2 dt,

and we consider the following problem. Given initial conditions (ξ1, ξ2) =(B, 1) with B > 1/2, where and when does the trajectory “land”, i.e. isit true that for some t > 0, (ξ1(t), ξ2(t)) = (C, 1), and what are C and t?Noting that for this system, we have

dξ2dξ1

=−D(2ξ1 − 1)

2ξ1,

we can integrate this to get

ξ2(ξ1) = Dη(ξ1) + C := D

(

ln(ξ1)

2− ξ1

)

+C,


and the solution which passes through (B, 1) is

ξ2(ξ1) = D(η(ξ1) − η(B)) + 1.

Since η is a function which is concave down with maximum at 1/2, thisshows that C < 1/2. Furthermore, this curve has maximum O(D). Thiswould seem to explain the bursting phenomenon well, since here we havea quick transition from (B, 1) to (C, 1) with an excursion height of O(D),which brings us to the start of another quiescent phase.

Unfortunately, this argument is not quite correct, because one can seethat the phase-locking assumption is not valid. Since the equation hasa transition into the ξ2 > Dξ1 region of phase-space (and in fact spendsmost of its time there), the locking in the θ equation switches signs, andθ now locks to π/4. This changes the signs in the dξj equations so makesthis analysis incorrect.

Now consider system (6.3), rescale ξ1 = x, ξ2 = Dy, τ = Dt, and writeα(t) = sin(2θ). The equations become

dx = 4αxy +1

D,

dy = −2y(2αx + 1),

dα = 2(y − x)(1 − α2),

(6.5)

and our initial condition is now (x0, y0) = (B, 1/D). The quantity αwants to lock to either 1 or −1 depending on the values of x and y. Letus split the positive quadrant into the two pieces R1 = {ξ1 > 1/2} andR2 = {ξ1 < 1/2}. By a simple phase plane analysis, one sees that forα = −1, the vector field points up and to the left in R1 and down and tothe right in R2. On the other hand, if α = 1, then the vector field pointsdown and to the right everywhere.

If we start at the point (B, 1/D), there are two possibilities of landingpoint (where we are “landing” at a point with y = 1/D). First, thetrajectory can move up and to the left, cross the line ξ1 = 1/2, andthen land at some point to the left of 1/2. This is the same picturewe would obtain assuming θ = 3π/4 throughout the burst. Second, thetrajectory can move to the right of the line x = 1/2 above y = 1, and thenland to the right of x = 1/2. But we will again be in the region whereξ2 ≪ Dξ1, and then the bursting process starts over again with this newinitial condition. The second case is a sort of “multibump” bursting. Inany case, the system will return quickly to the quiescent mode, no matterhow many bumps it traverses.


Furthermore, we claim that the multibump bursting is not importantfor large D dynamics. Recall that the initial condition B, while random,has mean less than 1. The probability of our having a burst which startsat, say, ξ1 = 10 is quite rare but as one can see from figure 6.3, for large Dthe burst has to start at ξ1 > 10 to get a multibump solution. Moreover,the probability of a multibump occurring goes to 0 as D → ∞.

−10 −8 −6 −4 −2 0 2 4−2

0

2

4

6

8

10

12

ln(x)

y

Figure 6.3. Evolution of system (6.5) for D = 104 and three initialconditions: (11, 1/D), (12, 1/D), (13, 1/D). The threshhold for getting amultibump solution seems to be somewhere between 12 and 13.

We can calculate the length of time the burst takes directly from thevector field. Take a (non-multibump) burst starting at (B, 1). Take B∗ =(B + 1/2)/2, and let C∗ be the solution to η(B∗) = η(C∗). Then thesolution will pass through the points (B, 1), (B∗, O(D)), (C∗, O(D)), and(C, 1). We can estimate how long it takes to pass though these points.To get from (B, 1) to (B∗, O(D)), the ξ2 variable must move an O(D)distance. In this region, dξ2 = O(Dξ2), so that it goes like eDt. It willthus take a ln(D)/D time to traverse this distance. The same argumentholds for the transit from C∗ to C. Now, to get from B∗ to C∗, notethat dξ1 = O(ξ1ξ2) = O(Dξ1), and it has to move a O(1) distance. Thiswill take O(1/D) time. All in all, the burst will take O(ln(D)/D) time.Compare this with figure 6.4, where the circles represent the proportion oftime the system spends in the bursting phase as a function of D. For thiscalculation, we took the system (6.1) with various D and initial condition(ξ1, ξ2, θ) = (1, 1, 0) and measured the proportion of time ξ2 > 1 fromt = 103 to t = 105.

In summary, the β = 0 case is a system which switches intermittentlybetween two quite distinct phases. It stays in the quiescent mode foran O(1) time, then bursts for an O(ln(D)/D) time, with a burst height


0 100 200 300 400 500 600 70010

−3

10−2

10−1

100

D

frac

tion

of ti

me

burs

ting

Figure 6.4. The solid curve is 0.7 ln(D)/D, and the circles are the pro-portion of time the system of (6.1) spends bursting as a function of D.

of O(D). From this, one can calculate how the moments of ξ2 act withrespect to D. During the quiescent phase, ξ2 moves between 0 and 1 likee±Dt. The pdf of ξ2 during this phase is f(ξ2) = (Dξ2)

−1. During thebursting phase, the solution moves from 1 toO(D) in time O(ln(D)/D), sothat the pdf in that phase is the uniform distribution ln(D)/D2. Thereforeif we were to calculate the moments of ξ2, we would have

〈ξn2 〉 =

∫ 1

0dξ2

ξn−12

D+

∫ D

1dξ2

ξn2 ln(D)

D2

= D−1 +Dn−1 lnD.

This shows that the two phases have quite different characters andexplains why there is no uniform rescaling which can describe the statisticsof both phases.

7 Summary

In this work, we have proposed a simple, two–mode model for nonlinearenergy transfer, where one mode receives a controlled amount of energythrough white noise, and the other dissipates energy through a nonlinearmechanism, which can be tuned to parameterize wave breaking.

The system exhibits a bifurcation between a quasi–equilibrium regimeand one far from equilibrium as an appropriate non-dimensional combi-nation of the driving and damping coefficients is increased. When thedamping is linear, a case previously studied in [9], the thermal regime isGaussian, and the non-equilibrium regime is highly intermittent. By con-trast, under nonlinear damping, the equilibrium regime has non–Gaussian


statistics, and the regime far from equilibrium is strongly constrained byslaving among the system’s degrees of freedom.

Acknowledgments

This work was partially supported by NSF via grants DMS01-01439,DMS02-09959, DMS02-39625, DMS-0417732, DMS-0306444, and by ONR.Some of the results of this paper appeared in the R.J.P.’s PhD thesis [10].

A Details of change of variables

We do the dξ2 integral in (5.11) by making the substitution ξ2 = E − ξ1.Noting that the dξ2 integral is over positive reals, this means that wemust have E − ξ1 ≥ 0, or 0 ≤ ξ1 ≤ E, and we get

∫ ∞

0dE

∫ E2/2

−E2/2dH

∫ 2π

0dθ

∫ E

0dξ1 δ(γ(ξ1)− H)F (H, E)φ(H, E, ξ1, E− ξ1, θ),

where γ(ξ1) = H = 2cos(2θ)(Eξ1 − ξ21). Let us for now consider theintegral only in the H > 0 regime. Thus we consider

∫ ∞

0dE

∫ E2/2

0dH

∫ π/4

−π/4dθ

∫ E

0dξ1 δ(γ(ξ1)−H)F (H, E)φ(H, E, ξ1, E−ξ1, θ).

We only need to consider those θ for which cos(2θ) > 0, as the sign ofH and the sign of cos(2θ) are the same. Let us note the following twosymmetries of the integrand: If we make the change θ 7→ −θ, every termin the integrand remains unchanged (every appearance of θ is as cos(2θ)),and if we make the change ξ1 7→ E − ξ1, the terms γ and F remainunchanged. We exploit these symmetries, and can write the innermosttwo integrals as

(A.1) 2

∫ π/4

0dθ

∫ E/2

0dξ1 δ(γ(ξ1) − H)F (H, E)(φ+ φ12),

where φ12 is φ with the roles of ξ1 and E − ξ1 interchanged. Now we dothe innermost integral, noting that γ is invertible on this domain. Usingthe rule that

∫ b

adx δ(γ(x) −A)f(x) =

f(γ−1(A))

γ′(γ−1(A))if A ∈ [γ(a), γ(b)],


we get the innermost integral to be

J1(E, H, cos(2θ))F (H, E)(φ+ φ12) if cos(2θ) ≥ 2H

E2,

where

J1(E, H, cos(2θ)) =1

2 cos(2θ)

√

E − 2Hcos(2θ)

,

and we use the substitution ξ1 7→ γ−1(H) = 12(E−

√

E2 − 2Hcos(2θ)). Chang-

ing variables in the dθ integral using ρ = cos(2θ) brings (A.1) to

(A.2)

∫ 2HE2

1dρ J2(E, H, ρ)F (φ+ φ12),

where

J2(H, E, ρ) =1

2ρ√

1 − ρ2

√

E2 − 2Hρ

,

and we make the further substitution in the integrand that cos(2θ) 7→ ρ.Substituting G = G(H,E) = G(2ξ1ξ2 cos(2θ), ξ1 + ξ2), (5.8) evaluates

to

φ = GE(1−2ξβ+2

2

2 ) +GH(−2ξ1ξβ+2

2

2 cos(2θ))

+GEEξ1 +GHE(4ξ1ξ2 cos(2θ)) +GHH(4ξ1ξ22).

(A.3)

If we denote

R± =1

2

E ±√

E2 − 2H

ρ

,

note that after the necessary substitutions we have

φ = GE(1 − 2Rβ+2

2

− ) +GH(−2R+Rβ+2

2

− ρ)

+GEER+ +GHE4R+R−ρ+GHH4R+R−R−,(A.4)

φ12 = GE(1 − 2Rβ+2

2

+ ) +GH(−2Rβ+2

2

+ R−ρ)

+GEER− +GHE4R−R+ρ+GHH4R+R+R−.(A.5)

So, our integrand is J2F (φ+ φ12), with

φ+ φ12 = GE(2 − 2(Rβ/2+1− +R

β/2+1+ )) +GH(−H(R

β/2− +R

β/2+ ))

EGEE + 4HGEH +2EH

ρGHH .


If we define

J(H, E) =

∫ 1

2H/E2

J2(H, E, ρ) dρ,

φ−1(H, E) =

∫ 1

2H/E2

ρ−1J2(H, E, ρ) dρ,

Sα(H, E) =

∫ 1

2H/E2

J2(H, E, ρ)(

R+(H, E, ρ)α +R−(H, E, ρ)α)

dρ,

ψ(H, E) =φ−1

J(H, E), Tα(H, E) =

Sα(H, E)

J(H, E)

(A.6)

then we see that (A.2) becomes, after dρ integration,

JF{(2 − 2Tβ/2+1)GE − HTβ/2GH

+ EGEE + 4HGEH + 2EHψGHH},

which proves (5.12). Also, note that T0(H, E) = 2 and T1(H, E) = E, sofor β = 0 this simplifies to

JF {2(1 − E)GE − 2HGH + EGEE + 4HGEH + 2EHψGHH} .

In general, however, Tα(H, E) is a function of both H and E, which meansthat for β > 0, the drift coefficients depend on both variables.

Bibliography

[1] D. J. Benney and A. C. Newell. Random wave closures. Stud. in Appl. Math.,1969.

[2] D. J. Benney and P. G. Saffman. Nonlinear interactions of random waves in adispersive medium. Proc. Roy. Soc. A, 1966.

[3] D. Cai, A. Majda, D. McLaughlin, and E. Tabak. Spectral bifurcations in disper-sive wave turbulence. Proc. Nat. Acad. Sci., 1999.

[4] D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak. Dispersive wave tur-bulence in one dimension. Phys. D, 152/153:551–572, 2001. Advances in nonlinearmathematics and science.

[5] A. D. D. Craik. Wave interactions and fluid flows. Cambridge Monographs onMechanics and Applied Mathematics. Cambridge University Press, Cambridge,1988.

[6] S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov. Optical tur-bulence: weak turbulence, condensates and collapsing filaments in the nonlinearSchrodinger equation. Phys. D, 57(1-2):96–160, 1992.


[7] K. Hasselmann. On the non-linear energy transfer in a gravity-wave spectrum. I.General theory. J. Fluid Mech., 12:481–500, 1962.

[8] A. J. Majda, D. W. McLaughlin, and E. G. Tabak. A one-dimensional model fordispersive wave turbulence. J. Nonlinear Sci., 7(1):9–44, 1997.

[9] P. A. Milewski, E. G. Tabak, and E. Vanden-Eijnden. Resonant wave interactionwith random forcing and dissipation. Stud. Appl. Math., 108(1):123–144, 2002.Dedicated to Professor David J. Benney.

[10] R. Pignol. Energy Transfer in Systems with Random forcing and nonlinear dissi-

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[11] V. E. Zakharov, V. L’vov, and G. Falkovich. Wave Turbulence. Springer, 1992.

R. E. Lee DeVille

Courant Institute251 Mercer StreetNew York, NY 10012E-mail: [email protected]

Paul A. Milewski

Department of Mathematics480 Lincoln DriveUniversity of WisconsinMadison, WI 10012E-mail: [email protected]

Ricardo J. Pignol

Departamento de MatematicaUniversidad Nacional del SurAvda. Alem 1253 (8000)Bahıa Blanca, ArgentinaEmail: [email protected]

Esteban G. Tabak


Eric Vanden-Eijnden


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